PhD thesis in English

2. Diagonalization of Transition Amplitudespotential brings about exponential localization along the main diagonal around itsminimum. The localization of dominant values of the transition amplitude to asmall area in the x − y plane gives practical justification for introduction of spacecutoff L in this approach.80706050t = 0.125t = 0.040t = 0.020t = 0.015E k403020100-100 5 10 15 20 25 30 35 40 45kFigure 2.2: Eigenspectrum of a free particle in a box. Eigenvalues E k are given asa function of level number k. The solid line gives the exact parabolic dispersionE k = π 2 (k + 1) 2 /8L 2 , while the dashed line presents results calculated in the tightbindingapproximation. The graph also shows numerical results obtained by thediagonalization of transition amplitudes for different values of time of evolution t.All the numerical calculations are for L = 6 and ∆ = 0.25, hence N cut = L/∆ = 24.In the continuum theory, the transition amplitude eigenproblem is mathematicallyequivalent to the Schrödinger equation. It is important to stress, however,that the procedure of space discretization introduces important differences betweeneigenproblems (2.3) and (2.6). In particular, as we will show in the next section, theprocedure based on the diagonalization of transition amplitudes leads to much faster(non-polynomial) convergence. An illustration of the relation of these two calculationschemes is shown in Fig. 2.2 which compares the exact parabolic dispersion ofa free particle in a box with numerical calculations based on diagonalizations of theHamiltonian and of the transition amplitudes. From the figure we see that the timeparameter t in the transition amplitude approach plays an important role. Increaseof the evolution time t gives better agreement with the exact dispersion relation.25